Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 225 / 3

The square-root distance between covariance operators isWriting $A_i=C_i^{1/2}$, minimizing ${\sum }_i\Vert A_i-A{\Vert }_{\mathrm{HS}}^2$ gives $A=n^{-1}{\sum }_i{A}_i$. Since this average is positive,the square-root barycenter of covariance operators.

Because $C_i=L_i{L}_i^{\ast }$ is a covariance operator,Thus both factors are Hilbert-Schmidt. The product of two Hilbert-Schmidt operators is a trace-class operator, with

The Procrustes distance between covariance operators isFor unitary $R$,The polar decomposition of a bounded operator impliesTherefore

The common eigenbasis givesConsequentlyFor the positive square-root factors, the singular values of $C_2^{1/2}{C}_1^{1/2}$ are $\sqrt{{\lambda }_k{\lambda }_k^{\ast }}$, soThus the distances coincide when the covariance operators commute and share an eigenbasis.